8. A patient is given a dosage Q of a drug at regular intervals of time T. The concentration of the drug in the blood has been shown experimentally to obey the law dC dt = a. If the first dose is administered at t = 0 hr, show that after T hr have elapsed, the residual R₁ = -kec where k is a positive constant. -In (kT+e-2) remains in the blood. b. Assuming an instantaneous rise in concentration whenever the drug is administered, show that after the second dose and T hr have elapsed again, the residual R₂ = − In [kT(1 + e¯2) +e=²0] remains in the blood. c. Show that the limiting value R of the residual concentrations for doses of Q mg/ml repeated at intervals of T hr is given by the formula R = - In kT 1 -e-Q d. Assuming the drug is ineffective below a concentration L and harmful above some higher concentration H, show that the dose schedule T for a safe and effective concentration of the drug in the blood satisfies the formula T 1 -L ` = = = =(e^² - e-H¹)
8. A patient is given a dosage Q of a drug at regular intervals of time T. The concentration of the drug in the blood has been shown experimentally to obey the law dC dt = a. If the first dose is administered at t = 0 hr, show that after T hr have elapsed, the residual R₁ = -kec where k is a positive constant. -In (kT+e-2) remains in the blood. b. Assuming an instantaneous rise in concentration whenever the drug is administered, show that after the second dose and T hr have elapsed again, the residual R₂ = − In [kT(1 + e¯2) +e=²0] remains in the blood. c. Show that the limiting value R of the residual concentrations for doses of Q mg/ml repeated at intervals of T hr is given by the formula R = - In kT 1 -e-Q d. Assuming the drug is ineffective below a concentration L and harmful above some higher concentration H, show that the dose schedule T for a safe and effective concentration of the drug in the blood satisfies the formula T 1 -L ` = = = =(e^² - e-H¹)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![8. A patient is given a dosage Q of a drug at regular intervals of time T. The concentration
of the drug in the blood has been shown experimentally to obey the law
dC
dt
=
-kec
a. If the first dose is administered at t = 0 hr, show that after T hr have elapsed, the
residual
R₁ = ln (kT+e=2)
remains in the blood.
b. Assuming an instantaneous rise in concentration whenever the drug is administered,
show that after the second dose and 7 hr have elapsed again, the residual
R₂ = − In [kT(1 + e¯²) +e=²²]
е
where k is a positive constant.
remains in the blood.
c. Show that the limiting value R of the residual concentrations for doses of Q mg/ml
repeated at intervals of T hr is given by the formula
R = - In
1
kT
-e-2
d. Assuming the drug is ineffective below a concentration L and harmful above some
higher concentration H, show that the dose schedule T for a safe and effective
concentration of the drug in the blood satisfies the formula
T = = 1/² (e-²- - e-")
k](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f71ceb2-3997-4087-a061-b3f0f635fe3e%2Fa861d199-47d5-46ad-9f09-fc9fbc033f6a%2Fqjvpgih_processed.png&w=3840&q=75)
Transcribed Image Text:8. A patient is given a dosage Q of a drug at regular intervals of time T. The concentration
of the drug in the blood has been shown experimentally to obey the law
dC
dt
=
-kec
a. If the first dose is administered at t = 0 hr, show that after T hr have elapsed, the
residual
R₁ = ln (kT+e=2)
remains in the blood.
b. Assuming an instantaneous rise in concentration whenever the drug is administered,
show that after the second dose and 7 hr have elapsed again, the residual
R₂ = − In [kT(1 + e¯²) +e=²²]
е
where k is a positive constant.
remains in the blood.
c. Show that the limiting value R of the residual concentrations for doses of Q mg/ml
repeated at intervals of T hr is given by the formula
R = - In
1
kT
-e-2
d. Assuming the drug is ineffective below a concentration L and harmful above some
higher concentration H, show that the dose schedule T for a safe and effective
concentration of the drug in the blood satisfies the formula
T = = 1/² (e-²- - e-")
k
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